Outline

1. ROTORCRAFT TRIM BY A NEURAL MODEL-PREDICTIVE AUTO-PILOTCarlo L. Bottasso and Luca RivielloPolitecnico di MilanoItaly31st European Rotorcraft Forum Firenze, Italy, 13-15 September 2005

2. Outline • Background and motivation: • Rotorcraft trim; • Possible solution strategies; • Non-linear Model-Predictive (NMP) auto-pilot: • Formulation; • Adaptive reduced model; • Numerical example; • Conclusions.

3. Introduction and Motivation Trim: control settings, attitude and cargo disposition for a desired steady (flight) condition. Performance, loads, noise, handling qualities, stability, etc. depend strongly on the trim condition. • Procedure: • Given desired loads or velocities specifying the desired condition, • Find resulting attitude and constant-in-time controls. TRIM PROBLEM • Important remark: • Rotorcraft systems excited by harmonic external loads; • Periodic response of all states and loads at trim.

4. Introduction and Motivation • Rotorcraft trim approaches: • Periodic shooting • Harmonic balance • Finite elements in time • Auto-pilot: • Adjust control settings to “fly” the system to the trimmed solution (Peters, Kim & Chen, 1984)(Peters, Chouchane & Fulton, 1990); • Very efficient even for large vehicle models (cost does not depend on the number of DOFs). Computational cost is a function of the number of DOFs.

5. Introduction and Motivation • High-fidelity comprehensive aeroelastic models: • Based on non-linear MB dynamics formulations; • Coupled with complex aerodynamic models or CFD. • Need for efficient trim procedures. • Current auto-pilots: • Are unsuitable for unstable systems; • Offer no guarantee of stability; • Often find limit cycle solutions. • Proposed approach: use non-linear model predictive (NMP) control technology for auto-pilot-based rotorcraft trim. Tilt-rotor whirl-flutter analysis (about 104 degrees of freedom)

6. e e e ¸ x u e e _ ( ) e e e f ¸ 0 x x u = ; ; ; ; _ ( ) e e e 0 c x x = ; ; Comprehensive Multibody Models • Comprehensive (multibody based) governing equations: • (dynamic & kinematic eqs.) • (constraints) • where: • System states :displacements/rotations, linear/angular velocities, internal states; • System controls :e.g. actuator inputs, controlled joint rotations, applied forces; • Lagrange multipliersthat enforce the constraints.

7. ¤ y T + t _ Z 1 ( ) ( ) e e e e e ¤ 8 8 8 T 0 t t t t t + + x y u y x z ; = = = ( ) e e e e d t ; ; ; ; : y g x u ; = ; T t Formulation of Rotorcraft Trim Problem • Define system outputs (problem dependent): • Wind tunnel trim: components of rotor loads in fixed system; • Free flight: capture gross vehicle motion. • Trim constraints: • where are desired values for the outputs; • Trim conditions: • Periodicity conditions: • (See Peters & Barwey 1996)

8. ¤ y : T + t T Z 1 ( ( ) ) e e e e µ µ µ u g x u ; = 0 1 1 ( ) e e e e d s c ³ ´ ³ ´ t ¼ ¼ ; ; ; : y g x u ; = ( ) µ Ã µ µ Ã µ Ã i i i i 1 2 3 4 ; + ¡ + ¡ T s n c o s = = i 0 1 1 s c ; ; ; ; : t 2 2 Rotorcraft Trim: Example Wind-tunnel trim: given advance ratio, find the controls that produce desired values of given average hub loads. • Hub loads: • Average hub loads: • Desired average hub loads: Blade pitch: Rotor controls:

9. e e e G ¤ u y y u ; ; ; f i ; ; · ¸ e e e e e e e @ ¡ ¡ ¡ 1 ¡ y y y y y y y ( ) e e e ¤ S G ¢ 1 0 2 0 0 t + ¡ n u u y y = S f i u ¼ ; = ; ; : : : ; : @ ¢ ¢ ¢ u 1 2 n u Trim Solution Strategies: Auto-pilot • Procedure: • Controls are promoted to dynamic variables; • Error on trim constraints is measured; • A suitable control law is designed to converge to the trim solution. • A possible proportional auto-pilot control law (in discrete form): • where: • Present/targetoutputs: - Initial/finalcontrols: • Gain matrix: • Input/output“sensitivity” matrix:

10. NMP Auto-pilot • Procedure: • Predictsystem response using a non-linear reduced model; • Compute controls tosteerthe system for a short time horizon; • Update reduced model based on predicted-actual output errors; • Iterate, shifting prediction forward (receding horizon control).

11. NMP Auto-pilot • Highlights: • Framework forguaranteeing stabilityof the closed-loop system; • Superior controlperformance (optimal control theory); • Constant-in-timeconstraintson controls explicitly enforced.

12. T Z ( ( ( ) ) [ ) ] e f ¤ ¤ ¤ T _ f 0 2 u p g y y y u y u y p g g ; = = i i i ( ) ¤ ; ; ; ; ; ; : d J M m a x m n t y y u = ( ( ) ) j j j j j j j j j j j j ¤ ¤ ; ; ; _ _ M T T T 0 · · t t ¡ + + < y y u u y y u u ; = = S S f i T ; ; ; ; c _ y u u i NMP Auto-pilot • Model-predictive tracking problem: solution yields steering controls . • Minimize cost • where • Subjected to: • Reduced model equations: • where is current estimate of model parameters. • Initial conditions: • Trim conditions: • Constraints: • Remark: constraints on controls (and states) are hard to incorporate in other control strategies.

13. e ¤ x u 0 e e _ ( ) e e ¤ f ¸ 0 x x u = h h h h ; ; ; ; _ ( ) e e e 0 c x x = h h ; ; t ( ) e e s e e r T x x = 0 0 : Steering Problem March forward in time multibody solver with given control inputs as computed by the tracking problem: Solve initial value problem from current state on steering window:

14. ( ) e e f ¤ _ 0 ¼ ¼ y u p y y u y u p = ; . ; ; ; Adaptive NMP Auto-pilot • Stability: guaranteed for infinite prediction horizon and reduced model identical to the plant. • Approximations: • Finite prediction horizon to lower computational cost; • Reduced model only approximates plant response. • Proposed solution: • Identify adaptive parametric reduced model to control the approximation error and converge to exact trim solution: • where the model parameters areoptimizedto have • when

15. e e d ¼ y u y u = . ( ) ( ) ( ( ) ) n f f d _ _ 0 y y u y y u y y u ; = = f f r e r e ; ; ; ; ; : : : ; ; ; Reduced Model • Reduced model: • Reference analytical model: • Reference model is problem dependent. • E.g.: wind tunnel trim  classical performance rotor model based on blade element theory with uniform inflow (Prouty 1990). • Augmented reduced model: • where is the unknown reference model defect that ensures • when

16. ( ) T T T T T i i i i i i j ( j ) ( ( ( ) ( ) ) ) d Á 8 Á b Á W V C C n 0 · > x p ¾ y a y u " " ¾ ¾ = = N 1 ; ; ; ; ; : : : ; ; : : : ; ; n T T T ( ( ) ) ( ) T T i i i i i i i i i i i i i i ( ( ( ) ) ) ( ( ( ) ) ) n n n d d d b b W V W V + + + p p y y y y u u p y ¾ x u a a " = = = = k k k k N N N N j j j j : : : ; ; : ; ; : : : : : : : : ; ; ; ; ; : : : ; ; : ; : : ; ; ; ; ; ; ; : : : : Reduced Model Identification Approximate with single-hidden-layer neural networks, one for each component: where and = reconstruction error (universal approximator, ); = matrices of synaptic weights and biases; = sigmoid activation functions; = network input. The reduced model parameters are readily identified with the synaptic weights and biases of the networks:

17. Numerical Example System • Wind-tunnel trim of a four-bladed flexible rotor: • UH-60 rotor multibody modelattached to the ground; • Three controls: bladecollectiveand longitudinal and lateral cyclicpitch angles; • Aerodynamics:strip theory. Reference model Analytical blade element/momentum theory, static flapping (performance model). Target Trim for three desired average hub load componentsin theinertial frame.

18. Numerical Example Finite element based MB code (Bauchau & Bottasso 2001).

19. ( ) k ( ) k e ¤ ¤ t t ¡ y y y " = s 2 s s Numerical Example Methodology • Given rotorcraft advance ratio (flight speed/tip speed) and weight,estimate the forces (ouputs)necessary to trim in straight level flight. • Then: • Initialize the controls to small values; • Activate the auto-pilot. Goal Steer rotor outputs to the desired values and evaluate controls in trim. Error definition where are (scaled) target trim outputs.

20. ( ) m a x m a x 8 T T 0 0 5 · ¸ t t : " " " = ; ; : : Numerical Example Time to trim: NPMA parameters: Activation freq.:4/rev; Prediction:3 rev; Num. of Neurons:20; Max. control rates:10 deg/sec. Classical auto-pilot stability limit. Dash-dotted: auto-pilot A; Dashed: auto-pilot B; Solid: NMPA.

21. ( ) m a x m a x 8 T T 0 0 1 · ¸ t t : " " " = ; ; : : Numerical Example Time to trim: NPMA parameters: Activation freq.:4/rev; Prediction:3 rev; Num. of Neurons:20; Max. control rates:10 deg/sec. Classical auto-pilot stability limit. Dash-dotted: auto-pilot A; Dashed: auto-pilot B; Solid: NMPA.

22. Numerical Example Controls: classic auto-pilot A, advance ratio 0.297.

23. Numerical Example Controls: NMP auto-pilot, advance ratio 0.297.

24. Numerical Example Outputs: classic auto-pilot A, advance ratio 0.297.

25. Numerical Example Outputs: NMP auto-pilot, advance ratio 0.297.

26. Numerical Example Outputs: classic auto-pilot A, advance ratio 0.297.

27. Numerical Example Outputs: NMP auto-pilot, advance ratio 0.297.

28. Conclusions • A new formulation for the auto-pilot approach was proposed, applicable to arbitrarily complex rotorcraft models; • Non-linear model predictive approach implies superior performance and leads to provable stability; • The solution specifically accounts for the presence of constant-in-time constraints on controls (trim conditions): no limit cycles; • Model adaptivity and learning reduce the need of tuning to different flight conditions and different models; • Extension to maneuvering flight: paper #29, Session C4, Flight Mechanics, Tue. 5:00-5:30.

29. Acknowledgements This work is supported in part by the US Army Research Office, through contract no. 99010 with the Georgia Institute of Technology, and a sub-contract with the Politecnico di Milano (Dr. Gary Anderson, technical monitor).